I am trying to reduce the computation of weak operadic colimits to colimits. Let me introduct some notation. Let $ q:C^{\otimes} \to N(Fin_*)$ be a

Let $ p: K \to C^{\otimes}_{act}$ be a diagram. Define $ p^{\otimes}: K \to C$ in the following way. For every $ k \in K$ , there is a cocartesian lifting $ s(k): p(k) \to k’$ of the unique active map $ qp(k) \to <1>$ .

Furthermore, we have an homotopy between $ qp: K \to N(Fin_*)^{act}$ and the constant functor $ <1>$ , because $ <1>$ is final in $ N(Fin_*)^{act}$ .

By a lemma used in the proof of the existence of transition functors, one can lift this homotopy to an homotopy $ H: K \times \Delta^1 \to C^{\otimes}_{act}$ . In particular, this yield a functor $ p^{\otimes}: K \to C$ , the underlying category of $ C^{\otimes}$ .

Set $ C^{act}_{p/}:= C \times_{C^{\otimes}_{act}} (C^{\otimes}_{act})_{p/} $ , i.e. cocones with cocone point lying over 1. I want to show that $ $ C^{act}_{p/} \simeq C_{p^{\otimes}/} $ $

The idea behind is the following: if $ K$ was a point, we could lift a map $ p^{\otimes}(k) \to c$ to a map $ p(k) \to c$ via the cocartesian lift $ s(k)$ . This realizes the intuitive idea that $ Mul_{C^{\otimes}}( (X_1, \ldots, X_n), c) \simeq Hom_C( X_1 \otimes \ldots \otimes X_n, c)$ . I believe that this could be generalized to arbitrary diagrams: intuitively, if we have a cone towards an object lying over 1, this factors through a cone lying over 1, made of “pointwise tensor products”.

Note that this would yield the following

**Corollary.** Let $ \bar{p}:K^{\triangleright} \to C^{\otimes}_{act}$ a diagram. Then it is a weak operadic q-colimit diagram iff $ \bar{p}^{\otimes}$ is a colimit diagram, because equivalences preserve initial objects.

I would be happy also if someone could solve the case $ K=\Delta^1$ , that would shed light on how to solve the general case simplex by simplex.